Message Boards Message Boards

Happy CPFC day! (common number of lifetime's prime-factors)

Fabian Wenger, algebra.se
Posted 4 years ago

Have some weekend fun figuring out when you share a day in life allowing a factorization that has an equal number of prime factors as other people around you. Take on the challenge to find the asymptotic probability that a day with common given prime factor count will appear in the future given the differences between birthdays and the calendar date you are interested in. Best regards, Fabian
POSTED BY: Fabian Wenger
5 Replies
Sort By:

enter image description here -- you have earned Featured Contributor Badge enter image description here Your exceptional post has been selected for our editorial column Staff Picks http://wolfr.am/StaffPicks and Your Profile is now distinguished by a Featured Contributor Badge and is displayed on the Featured Contributor Board. Thank you!
POSTED BY: EDITORIAL BOARD
Rohit Namjoshi
Rohit Namjoshi
Posted 4 years ago

Easier to read.

$\large\int_2^{\sqrt[3]{M}} -\frac{\log \left(\frac{\log (p)}{\log \left(\frac{M}{p^2}\right)}\right)}{p \log (p) \log \left(\frac{M}{p}\right)} \, dp$
POSTED BY: Rohit Namjoshi

Hint: The probability to randomly hit a day with a prime factor count of 2 if the number chosen is M is about Log[Log[M]/Log[2]-1]/Log[M]. There are more exact asymptotic forms but it may get rather messy. Here a numerical comparison for this approximate probability with a numerical one when averaging over the closest 10^4 integers as a function of M: 2 prime factor count probability asymptotics

For a 3-prime factor count I cannot get out a closed form expression but an integral that can be evaluated numerically:

\!\(
\*SubsuperscriptBox[\(\[Integral]\), \(2\), 
SuperscriptBox[\(M\), \(1/3\)]]\(\(-
\*FractionBox[\(Log[
\*FractionBox[\(Log[p]\), \(Log[
\*FractionBox[\(M\), 
SuperscriptBox[\(p\), \(2\)]]]\)]]\), \(p\ Log[
\*FractionBox[\(M\), \(p\)]]\ Log[p]\)]\) \[DifferentialD]p\)\)

and the numerical comparison is given in the following figure:

3 prime factor count asymptotics
POSTED BY: Fabian Wenger
Rohit Namjoshi
Rohit Namjoshi
Posted 4 years ago

Hi Fabian,

Cool idea! I have been trying it out on extended family members birthdates.

A more functional way to generate addDays.

addDays = 
 Subsets[bDates, {2}] // Map[-DateDifference[Sequence @@ ##] & /* QuantityMagnitude]
POSTED BY: Rohit Namjoshi

Thanks Rohit, always interesting to learn how to write better WL code, both Subsets and Sequence were new for me!
POSTED BY: Fabian Wenger
Reply to this discussion
Community posts can be styled and formatted using the Markdown syntax.
Tag limit exceeded
Note: Only the first five people you tag will receive an email notification; the other tagged names will appear as links to their profiles.
Reply Preview
Attachments
Remove
or Discard

Group Abstract Group Abstract